I was trying to find what is the probability of rolling 3 successes when rolling four d10 (ten-sided dice), with a success being any result of 7+, and I decided to use the AnyDice web app.
The result the app gave me was 17.92% for 3 successes and 52.48% for 2 successes. I then wanted to see how re-rolling 1 failure affected my chances of rolling 3 successes, and I tried using 2 different ways.
A) Since rolling again a die that showed a failure in the first roll gives a 40% for a success, I multiplied 52,48% * 0.4 to find the probability of rolling 3 successes with that second roll. This gives a ~21%.
B) For the second method, I decided to use AnyDice again, but this time calculate the probability of rolling 3 successes when rolling five d10. In my mind rolling 4 dice and re-rolling 1 failure was the same as rolling 5 dice and finding the 3 success % directly. This gave 31.74%, quite higher than the result of the first method.
So my questions are 1) Which result is the correct one? 2) If in the first method the re-rolled die is again a failure and I decide to roll it a third time, the probability of rolling 3 successes (accounting for action economy) is: 52.48% * 0.6 (for failing the 2nd roll) * 0.4 (for the 3rd roll) ~ 12.6%, right?
Thank you.